Legendrian barriers in prequantization bundles

Emmanuel Opshtein

REVIEW 2 major objections 4 minor 1 cited by

Legendrian barriers in prequantization bundles

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Prequantization bundles carry explicit Legendrian barriers: removing one makes long embedded contact flows impossible.

desk verdict Generalizes the S^3 Legendrian barrier to prequantization bundles, but the main theorem is proved only for quasi-holomorphic-section skeleta, not for all polarizations, and k=2 degenerates. read the letter →

arxiv 2512.18808 v2 pith:LHTZWM4T submitted 2025-12-21 math.SG

Emmanuel Opshtein This is my paper · ORCID

classification math.SG MSC 53D3553D1253D10

keywords LegendrianbarrierprequantizationbundleisotropicskeletoncontactHamiltonianLagrangiansymplecticdiscquasi-holomorphicsectionrigidity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that every prequantization bundle—the contact circle bundle over a closed symplectic manifold with integral symplectic class—contains explicit Legendrian barriers. For a polarization of degree $k\ge 2$, the barrier is the minimal Legendrian lift of the polarization's isotropic skeleton: it meets each fiber over the skeleton in $k$ equally spaced points. The main theorem says that any closed Legendrian submanifold moved by any contact Hamiltonian $H\ge 1$ either hits the barrier within time $1/k$, or its $1/k$-time sweep fails to be embedded. In other words, deleting the barrier destroys the possibility of long embedded Reeb or Hamiltonian cylinders. This gives a quantitative contact rigidity phenomenon in all dimensions and recovers the earlier three-dimensional sphere result as a special case.

What carries the argument

The proof's load-bearing object is a Liouville polarization of the negative symplectization, identified with a symplectic disc bundle over $N$. Starting from a quasi-holomorphic section $\sigma$ of the $k$th power of the tautological line bundle whose norm has non-degenerate critical points, the paper extends $\sigma$ to a compactifying projective sphere bundle via $s_\varepsilon=(1-R)^{k/2}P_R\sigma+\varepsilon R^{k/2}P_R\sigma_\infty$. The zero set of $s_\varepsilon$ is a smooth divisor, and the associated Liouville form has a skeleton equal to $\Phi^{-1}(\Lambda_k \times (0,1))$—the radial product of the Legendrian barrier, where $\Phi$ is the identification between the disc bundle and the negative symplectization. A decomposition theorem for symplectic disc bundles then identi

What would settle it

A direct test: in the standard prequantization $S^{2n+1} \to \mathbb{CP}^n$ with the degree-$2$ skeleton made of half great circles, look for a closed Legendrian $\Lambda$ and a contact Hamiltonian $H\ge 1$ such that $\Phi^t_H(\Lambda)$ stays embedded for $t\in [0,1/2]$ and avoids the two-point Legendrian lift $\Lambda_2$. Finding one would disprove the $1/k$ bound. Alternatively, exhibiting a degree-$2$ polarization whose skeleton is not realizable by any $\kappa < 1$ quasi-holomorphic section with non-degenerate norm logarithm would show the proof does not cover the theorem's stated hypotheses.

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Extended reading notes

Core claim

The central discovery is that the Legendrian lift of a Lagrangian barrier is itself a Legendrian barrier. Let $(N,\tau)$ be a closed symplectic manifold with integral symplectic class, $\Gamma_k$ an isotropic skeleton of a polarization of degree $k\ge 2$, and $\Lambda_k$ its minimal Legendrian lift to the prequantization bundle. For every closed Legendrian $\Lambda$ and every smooth contact Hamiltonian $H\ge 1$, either there is an $H$-chord from $\Lambda$ to $\Lambda_k$ of length at most $1/k$, or the union of $\Phi^t_H(\Lambda)$ over $t\in [0,1/k]$ is not embedded. When $H$ is autonomous this becomes a chord from $\Lambda$ to itself or to $\Lambda_k$ of length $\le 1/k$. A direct consequence is that no contact form $\alpha' \le \alpha$ admits a strong contact embedding of $D(1/k)^n \times S^1$ into $P\setminus \Lambda_k$.

Load-bearing premise

The theorem is stated for every polarization of degree $k\ge 2$, but the proof needs the skeleton to be the skeleton of a quasi-holomorphic section of the $k$th power line bundle with control constant $\kappa < 1$, transverse zeros, and non-degenerate critical points of the norm's logarithm; such sections are only known to exist for large $k$.

Editorial extensions

If this is right

For any contact form α'≤α, there is no strong contact embedding of D(1/k)^n×S^1 into P\Λ_k; the largest round cylinder that fits contactomorphically has radius at most 1/k.
Any Legendrian isotopy in P\Λ_k generated by a contact Hamiltonian H≥1 has discriminant length at least k, so the barrier forces every long isotopy to pass through non-embedded stages.
The area class of an isotropic skeleton of degree k is discrete with values in (1/k)Z; this rationality is exactly what makes the k-fold Legendrian lift well-defined.
For any smooth polarization of degree k, every Lagrangian immersion in M\L_k bounds a symplectic disc of area <1/k, so the Lagrangian capacity of the complement is at most 1/k.
For autonomous Hamiltonians the statement reduces to a chord from Λ to itself or to Λ_k of length ≤1/k, giving a concrete realization of the paper's notion of universal interlinking.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof requires the skeleton to be realizable by a quasi-holomorphic section with control constant κ<1; known existence results only guarantee such sections for large k. If some degree-2 polarization fails this condition, the theorem as stated would need a separate treatment for small k.
The same strategy, using a degenerate symplectic form on a projective compactification, may produce barriers in other open symplectic manifolds with rational symplectic class, not only negative symplectizations of prequantization bundles.
The paper's (δ,δ')-universal interlinker notion suggests that the constants 1/k could be sharp for contact capacities; testing whether δ and δ' can be tuned independently would give finer quantitative invariants.
The k=1 case appears to be genuinely exceptional—the statement already fails for complex projective space with the Fubini–Study form—so the rational-area mechanism underlying the proof seems to be the decisive feature.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here’s my take. The paper does something real: it upgrades the Legendrian-barrier phenomenon from S^3 [OS24] to every contact prequantization bundle over a closed integral symplectic manifold, with the constant 1/k built into the statement. That is not in the existing literature. The proof is a serious piece of work: a k-fold Legendrian lift construction (Cor 3.6), a degenerate non-symplectic compactification SPB, and a long explicit computation of the Liouville skeleton. The central dichotomy is plausible, and for k≥3 the computations mostly check out.

The soft spots are concentrated in the statement-versus-proof gap. Theorem 1 quantifies over all polarizations of degree k≥2. The proof, however, starts with a κ-quasi-holomorphic section σ of L^k, with κ<1 and ln|σ| Morse. The paper itself says such sections 'might exist when k≫1 by Donaldson–Giroux theory.' No bridge is given from an arbitrary polarization to such a section, and for small k (especially k=2) existence is not covered. So the theorem as stated is not proved.

The k=2 case is not a minor technicality. The critical-point radius formula contains the exponent 2/(k−2), which is undefined at k=2, and Lemma 5.4's Hessian coefficient −k(k−2) vanishes, so the Morse-index argument collapses. The paper gestures at taking ε large when k=2, but that does not repair the critical-point analysis. Thus the statement is not proved for k=2 even for quasi-holomorphic sections. This is a real gap, not a nitpick.

There are also several load-bearing citations to the author's own prior work ([Ops13], [Ops15]) and to SFT compactness without verification in the degenerate non-symplectic setting. That may be acceptable for a research article, but the reader should push for clarity.

Credit where due: the paper is honest about its assumptions — Section 5 explicitly flags the quasi-holomorphic existence issue. The algebraic core is written out in enough detail to be checked. The non-symplectic compactification is a clever move. If the polarization-to-section reduction is filled in (or the theorem statement narrowed) and the k=2 case is either repaired or excluded, the paper will be a strong contribution.

Who is this for? Anyone working on contact rigidity, Legendrian barriers, or quantitative contact topology. It deserves a serious referee. I would send it to review, with the expectation of major revision.

Referee Report

2 major / 4 minor

Summary. The paper studies Legendrian barriers in contact prequantization bundles. The main result, Theorem 1, asserts that for a closed symplectic manifold (N,τ) with integral symplectic class, any isotropic skeleton Γ_k of a polarization of degree k≥2 admits a k-fold Legendrian lift Λ_k in the prequantization bundle (P,α), and this lift is a Legendrian barrier: for any closed Legendrian Λ and any smooth contact Hamiltonian H≥1, either there is an H-chord from Λ to Λ_k of length ≤1/k, or the 1/k-sweep of Λ is not embedded. The proof uses Mohnke's trick to convert an embedded long sweep into a Lagrangian submanifold in the negative symplectization, then identifies the negative symplectization with a symplectic disc bundle over N. The technical core constructs, from a quasi-holomorphic section σ of L^k, a Liouville form whose skeleton is exactly the preimage of Λ_k×(0,1), and then applies Biran's decomposition and a ruled-manifold capacity bound. The paper also proves a Lagrangian analogue (Theorem 3) bounding the Cieliebak–Mohnke capacity of complements of polarizations. The proof is largely computational and self-contained, but there is a significant gap between the statement of Theorem 1 and the hypotheses actually used in its proof.

Significance. If the result holds in the stated generality, it is a substantial contribution: it generalizes the Legendrian barrier phenomenon from S^3 (proved in [OS24]) to all prequantization bundles, with the sharp constant 1/k and an explicit, geometrically meaningful barrier Λ_k. The construction via quasi-holomorphic sections and the degenerate compactification of the symplectic disc bundle are useful tools that may find further applications. The paper also gives a clean proof of a Lagrangian barrier theorem (Theorem 3) and provides a detailed account of Mohnke's trick. However, the central dichotomy is proved only for skeleta associated to quasi-holomorphic sections, while the theorem claims all degree-k polarizations. This mismatch, together with an apparent degeneration of the construction at k=2, prevents the main theorem from being established as stated.

major comments (2)

[Theorem 1 vs. Section 8] Theorem 1 quantifies over all isotropic skeleta associated to a polarization of degree k≥2 (Definition 3.4), but Section 8 begins with 'Let Γ_k ⊂ N be an isotropic skeleton associated to a quasi-holomorphic section of L^k → (N,τ).' No argument is given that every polarization degree k (especially k=2,3) admits a quasi-holomorphic section σ with transverse vanishing and ln|σ| Morse. Section 5 explicitly says such sections 'might exist when k≫1 by Donaldson-Giroux theory ... or for any other reason.' Since Proposition 7.4 and the subsequent Biran-decomposition step rely on the explicit s_ε construction, the proof covers only section-associated skeleta. The theorem should be restated in that restricted class, or a reduction from arbitrary polarizations to this class must be supplied.
[Section 5, k=2 degeneration] The critical-point analysis in Section 5 produces interior critical points at R = (1 + (ε/|σ(z)|)^{2/(k−2)})^{-1}. For k=2 this exponent is undefined; solving the critical-point equations forces |σ(z)|=ε, so for generic ε there are no interior critical points. Correspondingly, in Lemma 5.4 the Hessian term −k(k−2)ε^2 R^{k−3}/(2(1−R)) dR^2 vanishes, so −ln|s_ε| is not Morse in the radial direction, and Corollary 6.3 cannot identify stable manifolds as claimed. Lemma 6.5's later remark that k=2 can be handled by taking ε≫1 does not restore the missing critical points. Consequently Lemma 6.4 and Proposition 7.4 do not establish the skeleton equality Skel(s_ε)=Φ^{-1}(Λ_k×(0,1)) for k=2, so Theorem 1 is not proved in the stated k≥2 range.

minor comments (4)

[Abstract/Organization] The text 'theomem 1' in the Organization paragraph is a typo for 'Theorem 1'. Also 'Aknowledgements' should be 'Acknowledgements'.
[Notation] The notation L^k and L^{⊗k} is used interchangeably across Sections 3.5, 4.4, and 5; this may confuse readers. Consistency is recommended.
[Section 3.4] In the proof of Lemma 3.7, the unexplained symbol κ′(κ) is introduced but never used; either remove it or clarify its role.
[Definition 3.4] 'a s tuple' should read 'a tuple'. Also 'Poincaré-Dual' should be 'Poincaré dual' in several places.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; main risk is a scope gap between the stated theorem and the quasi-holomorphic-section proof, not circularity.

full rationale

Walking the derivation chain, I find no step in which a claimed prediction is equal, by construction or by a fitted parameter, to an input. The barrier constant 1/k is the degree of the polarization / residue of the Liouville form constructed in Section 7, not a fitted value; Proposition 3.5 derives the k-equipartition from the area class (1/k)Z, and the main estimate T = A_min(L) < 1/k comes from Theorem 3.1 after Biran-compactifying the basin supplied by Proposition 7.4. Self-citations are present but not circularly load-bearing: [OS24] is the motivating S^3 theorem, [Ops13] is used as an independent theorem identifying basins of tame Liouville forms with disc bundles, and [Ops15] supplies a residue computation; none of these assumes the conclusion of Theorem 1. The real weakness is a statement/proof mismatch: Theorem 1 quantifies over all polarizations of degree k ≥ 2, while Section 8 opens with “an isotropic skeleton associated to a quasi-holomorphic section”, and Section 5 concedes “Such sections might exist when k≫1 by Donaldson-Giroux theory”. The k = 2 case also degenerates in the displayed critical-radius formula (exponent 2/(k−2) and the vanishing −k(k−2) coefficient in Lemma 5.4). These are unproved generality and technical gaps, not circular reductions: an unsupported bridge from polarizations to quasi-holomorphic sections does not make the theorem's output an input. Score 2 reflects the minor self-citations; no circular step is scored.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No speculative entities are introduced: Lambda_k and Z_k are constructed from the input data (polarization/section and degree k), not postulated. Free parameters are limited to the deformation parameter epsilon and the implicit choice of quasi-holomorphic section; constants 1/k and the fiber area are structural inputs, not fit parameters. The main external inputs are the SFT compactness toolkit, Donaldson-Giroux quasi-holomorphic sections (only asymptotic in k), the author's own [Ops13] and [Ops15], and standard Gromov/positivity theory.

free parameters (1)

epsilon (deformation parameter in s_epsilon = s_0 + epsilon s_infinity) = epsilon > 2 max_N |sigma| when k = 2; small for k >= 3
Hand-chosen deformation parameter controlling the zero set Z(s_epsilon) and the skeleton. The final bound 1/k is independent of epsilon, but the construction requires epsilon in a specific range, qualitatively different for k = 2 (Lemma 6.5).

assumptions (6)

domain assumption SFT compactness (Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder) applies to the stretched-neck degenerations used in Theorem 3.1
Invoked in the proof of Lemma 3.2/Theorem 3.1 to produce the holomorphic building B with a sphere in the fiber class through p; standard but heavy machinery, assumed rather than proved.
domain assumption Existence of a kappa-quasi-holomorphic section sigma of L^k, transverse to zero, with ln|sigma| Morse and kappa < 1
Opening of Section 5. Guaranteed by Donaldson-Giroux only for k >> 1; for small k (in particular k = 2) the input is an assumption not delivered by the cited theory, while Theorem 1 is stated for all polarizations of degree k >= 2.
domain assumption The basin of attraction of a tame Liouville form with residue -1/k is a symplectic disc bundle SDB(Z_k, 1/k)
Used in Section 8 via the author's own [Ops13]. A published external result, but self-cited, and the degenerate context of Section 4.3 (non-symplectic compactification) is not identical to the standard setting.
domain assumption The Liouville form lambda on N\Sigma has residue -1/k along Sigma (rationality of Lagrangian skeleta)
Proposition 3.5; the proof is sketched and completed by reference to the author's own [Ops15]. Load-bearing for Corollary 3.6 (holonomy in Z_k) and for the kZ arithmetic in Section 8.
standard math Positivity of intersections and Gromov compactness for the fiber class F in the sphere bundle
Used in Lemma 3.2's compactness argument (only one bubble in class F, no H_2(Hur)(Y) bubbles); standard symplectic-topology input.
domain assumption Morse-theory dichotomy: for the gradient-like flow X_epsilon, every forward trajectory either reaches Z(s_epsilon) in finite time or converges to a critical point of phi_epsilon
End of Section 6: 'classical arguments of Morse theory', after Lemma 6.5 shows X_epsilon points away from N_infinity and is tangent to N_0. Requires Lyapunov/properness control of phi_epsilon = -ln|s_epsilon| on the open manifold SDB(N, tau).

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Cite this review

Pith. "Pith review of Legendrian barriers in prequantization bundles." pith.science (2026). https://pith.science/paper/LHTZWM4T

@misc{pith2026251218808,
  author       = {Pith},
  title        = {Pith review of: Legendrian barriers in prequantization bundles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHTZWM4T}},
  note         = {Machine review of arXiv:2512.18808}
}

read the original abstract

We show that prequantization bundles have explicit Legendrian barriers, whose removal obstruct the embedding of long cylinders over Legendrian submanifolds.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Lengths of Reeb chords and Viterbo restriction
math.SG 2026-06 accept novelty 6.5 of 10

Products of spheres yield n-invertible cotangent bundles, so every compact Legendrian in ST*M has a Reeb chord of length at most the n-invertibility capacity.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Spectral selectors and contact orderability, 2025

Simon Allais and Pierre-Alexandre Arlove. Spectral selectors and contact orderability, 2025

2025

[2] [2]

Bourgeois, Y

F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder. Compactness results in symplectic field theory. Geom. Topol. , 7:799--888, 2003

2003

[3] [3]

A stability property of symplectic packing

Paul Biran. A stability property of symplectic packing. Invent. Math. , 136(1):123--155, 1999

1999

[4] [4]

P. Biran. Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal. , 11(3):407--464, 2001

2001

[5] [5]

Contact non-squeezing and orderability via the shape invariant

Dylan Cant. Contact non-squeezing and orderability via the shape invariant. Internat. J. Math. , 35(3):Paper No. 2350109, 6, 2024

2024

[6] [6]

Cieliebak and K

K. Cieliebak and K. Mohnke. Punctured holomorphic curves and L agrangian embeddings. Invent. Math. , 212(1):213--295, 2018

2018

[7] [7]

The discriminant and oscillation lengths for contact and L egendrian isotopies

Vincent Colin and Sheila Sandon. The discriminant and oscillation lengths for contact and L egendrian isotopies. J. Eur. Math. Soc. (JEMS) , 17(7):1657--1685, 2015

2015

[8] [8]

S. K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Differential Geom. , 44(4):666--705, 1996

1996